Subgroups — #30875593

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22563174	C₂	1	2	yes	maximal
2	#22561071	C₂	1	2	yes	maximal
3	#22560845	C₂	1	4	yes	maximal
4	#22560373	C₂	1	1	yes	maximal, characteristic
5	#22560224	C₂	1	2	yes	maximal
6	#107×Dic₆₇	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523977	C₂²	1	2	yes	—
10	C₂×#14523975	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14523967	C₂²	1	4	yes	—
13	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
14	#16863303	C₂²	1	4	yes	—
15	#16863282	—	2	4	no	—
16	#16863264	—	2	2	no	—
17	#16863262	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
20	#16863242	—	2	2	no	—
21	#16863158	C₂²	1	4	yes	—
22	#16863150	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₇	C₄	1	8	yes	complemented
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
32	C₂×C₄×Dic₆₇	—	2	4	no	—
33	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
34	#14523977	—	2	4	no	—
35	#14523975	—	4	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	#14523967	—	4	4	no	—
38	C₈×Dic₆₇	—	2	4	no	—
39	C₆₇⋊#95	—	2	4	no	—
40	C₆₇⋊#95	—	2	4	no	—
41	C₆₇⋊#95	—	2	4	no	—
42	C₆₇⋊#95	—	2	2	no	—
43	C₆₇⋊#95	—	2	2	no	—
44	C₆₇⋊#95	D₄	1	1	yes	characteristic
45	C₆₇⋊#95	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	#107×C₆₇	C₂×C₄	1	4	yes	—
59	#103×C₆₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
61	C₂²×Dic₆₇	—	2	4	no	—
62	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
63	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	—	2	2	no	—
72	C₄×Dic₆₇	—	4	8	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₆₇	—	2	8	no	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	#41	1	2	yes	—
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₄⋊C₄×C₆₇	—	2	8	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
94	C₄²×C₆₇	—	2	8	no	—
95	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₆₇	#41	1	2	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₆₇	—	2	4	no	—
99	Q₈×C₆₇	—	2	4	no	—
100	Q₈×C₆₇	—	4	2	no	—
101	Q₈×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₆₇	—	2	2	no	—
118	C₂×Dic₆₇	—	2	2	no	—
119	C₂×Dic₆₇	—	2	2	no	—
120	C₂×Dic₆₇	—	2	2	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	—	4	4	no	—
124	C₂×Dic₆₇	—	4	4	no	—
125	C₂×Dic₆₇	#135	1	4	yes	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	#117	1	4	yes	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₆₇	#547	1	1	yes	characteristic
131	C₂²×C₆₇	#550	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×#135	1	2	yes	—
133	C₂²×C₆₇	#578	1	2	yes	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	—	4	4	no	—
138	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
139	C₄×C₆₇	#520	1	1	yes	characteristic
140	C₄×C₆₇	#555	1	1	yes	characteristic
141	C₄×C₆₇	#556	1	1	yes	characteristic
142	Dic₆₇	—	2	4	no	—
143	Dic₆₇	—	4	2	no	—
144	Dic₆₇	—	8	4	no	—
145	#22809	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	#2354	1	1	yes	characteristic
147	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
148	C₂×C₆₇	#2950	1	1	yes	characteristic
149	C₂×C₆₇	#3336	1	2	yes	—
150	C₂×C₆₇	#3393	1	2	yes	—
151	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
152	#2356	—	67	2	no	—
153	#1969	—	67	2	no	—
154	#1927	—	67	4	no	—
155	#1846	—	67	1	no	—
156	#1805	—	67	2	no	—
157	C₄×#107	—	67	2	no	—
158	C₈×C₄⋊C₄	—	67	1	no	—
159	C₆₇	#22809	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
161	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
162	C₄×Q₁₆	Dic₆₇	1	8	yes	complemented
163	C₂×#107	—	67	2	no	—
164	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#106	—	67	2	no	—
166	C₂×C₄⋊C₈	—	67	1	no	—
167	C₂×#103	—	67	4	no	—
168	C₂×#103	C₂×D₆₇	1	2	yes	—
169	C₂×C₄×C₈	—	67	2	no	—
170	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
171	#391	—	67	4	no	—
172	#387	—	134	4	no	—
173	#383	—	134	2	no	—
174	#382	—	67	2	no	—
175	C₄×C₄⋊C₄	—	67	1	no	—
176	C₄×C₄⋊C₄	—	67	4	no	—
177	#376	—	134	2	no	—
178	#340	—	67	4	no	—
179	#336	—	67	2	no	—
180	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
184	C₂²×C₈	—	67	2	no	—
185	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	67	2	no	—
191	C₂×C₄⋊C₄	—	67	4	no	—
192	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	67	2	no	—
195	C₂×C₄²	—	134	4	no	—
196	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
197	#107	—	134	4	no	—
198	#107	C₂×Dic₆₇	1	4	yes	—
199	#106	—	268	2	no	—
200	C₄⋊C₈	—	134	2	no	—
201	#103	—	268	4	no	—
202	#103	C₂×Dic₆₇	1	8	yes	—
203	C₄×C₈	—	134	4	no	—
204	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
205	#95	—	67	1	no	—
206	#95	—	67	1	no	—
207	#95	—	134	4	no	—
208	#95	—	134	4	no	—
209	#95	—	134	4	no	—
210	#95	—	134	2	no	—
211	#95	—	134	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	67	2	no	—
221	C₂²×C₄	—	67	2	no	—
222	C₂²×C₄	—	134	4	no	—
223	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	—	134	2	no	—
232	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
233	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
234	C₂×C₈	#11761628	1	2	yes	—
235	C₂×C₈	#11761628	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	134	4	no	—
239	C₄⋊C₄	—	134	4	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	268	8	no	—
244	C₄²	—	134	2	no	—
245	C₄²	—	134	2	no	—
246	C₄²	C₂²×Dic₆₇	1	2	yes	—
247	C₄²	#11761627	1	2	yes	—
248	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	268	4	no	—
261	C₂×C₄	—	268	4	no	—
262	C₂×C₄	—	268	4	no	—
263	C₂×C₄	—	67	4	no	—
264	C₂×C₄	—	134	2	no	—
265	C₂×C₄	—	134	2	no	—
266	C₂×C₄	—	134	2	no	—
267	C₂×C₄	—	134	2	no	—
268	C₂×C₄	—	134	2	no	—
269	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
270	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
271	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
273	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
274	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
275	C₂×C₄	#14524089	1	1	yes	characteristic
276	C₂×C₄	#14524090	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	268	2	no	—
279	C₈	#14524002	1	4	yes	—
280	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
281	C₂²	#16864051	1	1	yes	characteristic
282	C₂²	#16864067	1	1	yes	characteristic
283	C₂²	#135×D₆₇	1	2	yes	—
284	C₂²	#16864209	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	268	2	no	—
290	C₄	—	536	4	no	—
291	C₄	—	134	4	no	—
292	C₄	C₂×#14524002	1	1	yes	characteristic
293	C₄	#16863879	1	1	yes	characteristic
294	C₄	#16864111	1	1	yes	characteristic
295	C₄	#16864116	1	1	yes	characteristic
296	C₂	#22563158	1	1	yes	characteristic
297	C₂	#135×Dic₆₇	1	1	yes	characteristic
298	C₂	#22569077	1	1	yes	characteristic
299	C₂	#22572579	1	2	yes	—
300	C₂	#22573197	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#22809
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
25. C₄×Q₁₆×C₆₇	C₄	semidirect
159. C₆₇	#22809	semidirect
162. C₄×Q₁₆	Dic₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22563174	2
#22561071	2
#22560845	2
#22560373	2
#22560224	2
#107×Dic₆₇	2
C₈×C₄⋊Dic₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
#22809	67